Abstract
In this paper, a new subspace gradient method is proposed in which the search direction is determined by solving an approximate quadratic model in which a simple symmetric matrix is used to estimate the Hessian matrix in a three-dimensional subspace. The obtained algorithm has the ability to automatically adjust the search direction according to the feedback from experiments. Under some mild assumptions, we use the generalized line search with non-monotonicity to obtain remarkable results, which not only establishes the global convergence of the algorithm for general functions, but also R-linear convergence for uniformly convex functions is further proved. The numerical performance for both the traditional test functions and image restoration problems show that the proposed algorithm is efficient.
Highlights
The Conjugate Gradient (CG) method is dedicated to solving the unconstrained optimization problem: min f (x), (1)x∈ n where f : Rn → R is smooth and the gradient of f (x) at xk is marked gk
We focus on the following generalized line search, which has been shown to be very efficient for CG methods in [11]
We report the numerical performance of the DSCG algorithm from two aspects
Summary
The Conjugate Gradient (CG) method is dedicated to solving the unconstrained optimization problem: min f (x), (1). X∈ n where f : Rn → R is smooth and the gradient of f (x) at xk is marked gk. The advantages of its simple form and low storage requirements make the CG method a powerful tool for dealing with problem (1). It starts at a starting point x0 and generates an iterative sequence. That is, xk moves forward by one step αk along the search direction dk and reaches the (k + 1)-th iteration point xk+1. The direction dk is usually defined as dk = −gk, if k = 0,. More relevant research and the progress of CG method can be found in the literature [7,8,9,10]
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